Forexampleifa080GeVkaonbeamisincidentonaprotontargetthecenterofmassenergyis1699GeVandthecenterofmassmomentumofeitherparticleis0442GeVItisalsousefultonotethat1lab1lablabCITATIONSEidelmanetal ID: 514006
Download Pdf The PPT/PDF document "38.Kinematics38.KINEMATICSRevisedJanuary..." is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
38.Kinematics38.KINEMATICSRevisedJanuary2000byJ.D.Jackson(LBNL).Throughoutthissectionunitsareusedinwhich=1.Thefollowingconversionsareuseful:=197.3MeVfm,(=0.3894(GeV)mb.38.1.LorentztransformationsTheenergyand3-momentumofaparticleofmassforma4-vectorwhosesquare.Thevelocityoftheparticleis.Theenergyandmomentum()viewedfromaframemovingwithvelocityaregivenby=(1and)arethecomponentsofperpendicular(parallel)to.Other4-vectors,suchasthespace-timecoordinatesofevents,ofcoursetransforminthesameway.Thescalarproductoftwo4-momentaisinvariant(frameindependent).38.2.Center-of-massenergyandmomentumInthecollisionoftwoparticlesofmassesandthetotalcenter-of-massenergycanbeexpressedintheLorentz-invariantformistheanglebetweentheparticles.Intheframewhereoneparticle(ofmassisatrest(labframe),1labThevelocityofthecenter-of-massinthelabframeislab1lablab1laband1labThec.m.momentaofparticles1and2areofmagnitudelab Forexample,ifa0.80GeV/kaonbeamisincidentonaprotontarget,thecenterofmassenergyis1.699GeVandthecenterofmassmomentumofeitherparticleis0.442GeV/Itisalsousefultonotethat1lab1lablabCITATION:S.Eidelmanetal.,PhysicsLettersB592,1(2004)availableonthePDGWWWpages(URL:)January4,200512:11 38.Kinematics38.3.Lorentz-invariantamplitudesThematrixelementsforascatteringordecayprocessarewrittenintermsofaninvariantamplitude.Asanexample,the-matrixfor22scatteringisrelated Thestatenormalizationissuchthat=(238.4.ParticledecaysThepartialdecayrateofaparticleofmassintobodiesinitsrestframeisgivenintermsoftheLorentz-invariantmatrixelement ;:::;pisanelementof-bodyphasespacegivenby;:::;p Thisphasespacecanbegeneratedrecursively,viz.;:::;p;:::;pq;p;:::;p.Thisformisparticularlyusefulinthecasewhereaparticledecaysintoanotherparticlethatsubsequentlydecays.Survivalprobability:Ifaparticleofmasshasmeanproperlifetime(=1)andhasmomentum(),thentheprobabilitythatitlivesforatimegreaterbeforedecayingisgivenbyandtheprobabilitythatittravelsadistanceorgreaterisJanuary4,200512:11 38.Kinematics Figure38.1:Denitionsofvariablesfortwo-bodydecays.Two-bodydecaysIntherestframeofaparticleofmass,decayinginto2particleslabeled1and2, 2M;(38:15)jp1j=jp2j=M2(m1+m2)2M2(m1m2)21=2 and 2jMj2jp1j (cos)isthesolidangleofparticle1.Three-bodydecays Figure38.2:Denitionsofvariablesforthree-bodydecays.Deningand,thenand,whereistheenergyofparticle3intherestframeof.Inthatframe,themomentaofthethreedecayparticleslieinaplane.Therelativeorientationofthesethreemomentaisxediftheirenergiesareknown.ThemomentacanthereforebespeciedinspacebygivingthreeEulerangles(;; )thatspecifytheorientationofthenalsystemrelativetotheinitialparticle[1].Then )51 dd(cosd :January4,200512:11 38.KinematicsAlternatively )51 where()isthemomentumofparticle1intherestframeof1and2,and istheangleofparticle3intherestframeofthedecayingparticle.andaregivenby and [ComparewithEq.(38Ifthedecayingparticleisascalarorweaverageoveritsspinstates,thenintegrationovertheanglesinEq.(3818)gives )31 8M jMj21dE2=1 )31 32M3 ThisisthestandardformfortheDalitzplot.Dalitzplot:Foragivenvalueof,therangeofisdeterminedbyitsvalueswhenisparallelorantiparalleltomax E22m22q E23m232;:22a)(m2)=(E2+E3)2q E22m22+q Hereandaretheenergiesofparticles2and3intherestframe.ThescatterplotinandiscalledaDalitzplot.If isconstant,theallowedregionoftheplotwillbeuniformlypopulatedwithevents[seeEq.(3821)].Anonuniformityintheplotgivesimmediateinformation.Forexample,inthecaseofK,bandsappearwhenre ectingtheappearanceofthedecaychainKJanuary4,200512:11 38.Kinematics 012345 10 m12 (GeV2)m23 (GeV2) (m1+m2)2(M-m3)2 (M-m1)2(m2+m3)2 (m23)min Figure38.3:Dalitzplotforathree-bodynalstate.Inthisexample,thestate at3GeV.Four-momentumconservationrestrictseventstotheshadedregion.Kinematiclimits:Inathree-bodydecaythemaximumof,[givenbyEq.(3820)],isachievedwhen,particles1and2havethesamevectorvelocityintherestframeofthedecayingparticle.If,inaddition,thenmaxmaxmaxMultibodydecays:Theaboveresultsmaybegeneralizedtonalstatescontaininganynumberofparticlesbycombiningsomeoftheparticlesinto\eectiveparticles"andtreatingthenalstatesas2or3\eectiveparticle"states.Thus,ifijk:::,thenijk::: ijk:::andijk:::maybeusedinplaceofintherelationsinSec.38.4.3or38.4.3.1above. Figure38.4:Denitionsofvariablesforproductionofan-bodynalstate.January4,200512:11 38.Kinematics38.5.CrosssectionsThedierentialcrosssectionisgivenby 4q ;:::;p[SeeEq.(3811).]Intherestframeof(lab), 1lab;(38whileinthecenter-of-massframe 1cm Two-bodyreactions Figure38.5:Denitionsofvariablesforatwo-bodynalstate.Twoparticlesofmomentaandandmassesandscattertoparticlesofmomentaandandmassesand;theLorentz-invariantMandelstamvariablesaredenedbyandtheysatisfyThetwo-bodycrosssectionmaybewrittenas dt=1 1 1cmJanuary4,200512:11 38.KinematicsInthecenter-of-massframe1cm3cm1cm3cm1cm3cmsin1cm3cmsinistheanglebetweenparticle1and3.Thelimitingvalues=0)and)for22scatteringare 2p 1cm3cmIntheliteraturethenotationmax)for)issometimesused,whichshouldbediscouragedsince.Thecenter-of-massenergiesandmomentaoftheincomingparticlesare1cm 2p 2cm 2p For3cmand4cm,changeand.Then and1cm1lab p Herethesubscriptlabreferstotheframewhereparticle2isatrest.[ForotherrelationsseeEqs.(38.2){(38.4).]Inclusivereactions:Choosesomedirection(usuallythebeamdirection)forthe-axis;thentheenergyandmomentumofaparticlecanbewrittenasy;psinhisthetransversemassandtherapidityisdenedby 2E+pz =ln =tanh Underaboostinthe-directiontoaframewithvelocitytanh.HencetheshapeoftherapiditydistributiondN=dyisinvariant.Theinvariantcrosssectionmayalsoberewritten d3p=d3 ddyp dydJanuary4,200512:11 38.KinematicsThesecondformisobtainedusingtheidentitydy=dp,andthethirdformrepresentstheaverageoverFeynman'svariableisgivenby max maxInthec.m.frame, p sinh p andmax=ln( For,therapidity[Eq.(3837)]maybeexpandedtoobtain 2)+ sin2)+lntan(wherecos.Thepseudorapiditydenedbythesecondlineisapproximatelyequaltotherapidityforand,andinanycasecanbemeasuredwhenthemassandmomentumoftheparticleisunknown.Fromthedenitiononecanobtaintheidentitiessinh=cotsintanh=cosPartialwaves:TheamplitudeinthecenterofmassforelasticscatteringofspinlessparticlesmaybeexpandedinLegendrepolynomialsk; +1)(cosisthec.m.momentum,isthec.m.scatteringangle,1,andisthephaseshiftofthepartialwave.Forpurelyelasticscattering,=1.Thedierentialcrosssectionis k;Theopticaltheoremstatesthat andthecrosssectioninthepartialwaveisthereforebounded: +1)+1) January4,200512:11 38.Kinematics 1/21/2 al d Figure38.6:Argandplotshowingapartial-waveamplitudeasafunctionofenergy.Theamplitudeleavestheunitarycirclewhereinelasticitysetsin(Theevolutionwithenergyofapartial-waveamplitudecanbedisplayedasatrajectoryinanArgandplot,asshowninFig.38.6.TheusualLorentz-invariantmatrixelement(seeSec.38.3above)fortheelasticprocessisrelatedtok;)by k; lab=0)andarethecenter-of-massenergysquaredandmomentumtransfersquared,respectively(seeSec.38.4.1).Resonances:TheBreit-Wigner(nonrelativistic)formforanelasticamplitudewitharesonanceatc.m.energy,elasticwidth,andtotalwidth isthec.m.energy.AsshowninFig.38.7,intheabsenceofbackgroundtheelasticamplitudetracesacounterclockwisecirclewithcenter2andradiuswheretheelasticity.TheamplitudehasapoleatThespin-averagedBreit-Wignercrosssectionforaspin-resonanceproducedinthecollisionofparticlesofspinand+1) +1)(2+1) k2BB2tot isthec.m.momentum,isthec.m.energy,andandarethebranchingfractionsoftheresonanceintotheentranceandexitchannels.The2factorsarethemultiplicitiesoftheincidentspinstates,andarereplacedby2forphotons.Thisexpressionisvalidonlyforanisolatedstate.Ifthewidthisnotsmall,cannotbetreatedasaconstantindependentof.Therearemanyotherformsfor,allofwhichareequivalenttotheonegivenhereinthenarrow-widthcase.Someoftheseformsmaybemoreappropriateiftheresonanceisbroad.January4,200512:11 38.Kinematics 1/21/2 el/2 Figure38.7:Argandplotforaresonance.TherelativisticBreit-WignerformcorrespondingtoEq.(3850)is: Abetterformincorporatestheknownkinematicdependences,replacing ),where)isthewidththeresonanceparticlewouldhaveifitsmasswere ,andcorrespondingly )where)isthepartialwidthintheincidentchannelforamass s:a`=p s(s) sm2+ip Fortheboson,allthedecaysaretoparticleswhosemassesaresmallenoughtobeignored,soondimensionalgrounds ,wheredenesthewidthofthe,and)isconstant.Afulltreatmentofthelineshaperequiresconsiderationofdynamics,notjustkinematics.ForthethisisdonebycalculatingtheradiativecorrectionsintheStandardModel.1.See,forexample,J.J.Sakurai,ModernQuantumMechnaics,Addison-Wesley(1985),p.172,orD.M.BrinkandG.R.Satchler,AngularMomentum,2nded.,OxfordUniversityPress(1968),p.20.January4,200512:11